91Ó°ÊÓ

Given table is

and, $\mathrm{Î\pm }=0.01$. we have to find out whether the data provide sufficient evidence to conclude that the predictor variable is useful for providing the response variable.

STEP -1: The null and alternative hypothesis are

$H_0:{\mathrm{β}}_1=0$

$H_{\mathrm{Î\pm }}:{\mathrm{β}}_1≠0$

STEP 2: Determine the$\left(\mathrm{Î\pm }\right)$significance level.

The hypothesis test should be run at a significance level of $1$percent, or $\mathrm{Î\pm }=0.01$

Table of computations:

$$\begin{gathered} S_{xy}=∑x_i{y}_i-\left(∑x_i\right)\left(∑y_i\right)/n \\ =169993-\left(270\right)\left(5552\right)/9 \\ =3433 \end{gathered}$$

$$\begin{gathered} S_{xx}=∑x_i^2-{\left(∑x_i\right)}^2/n \\ =8316-(270{)}^2/9 \\ =216 \end{gathered}$$

The total sum of square SST is calculated by

$$\begin{gathered} S_{yy}=∑y_i^2-{\left(∑y_i\right)}^2/n \\ =3504412-(5552{)}^2/9 \\ =3504412-30824704/9 \\ =79444.88889 \end{gathered}$$

The regression sum of square is calculated by

$$\begin{gathered} SSR=\frac{S_{xy}^2}{S_{xx}} \\ =\frac{(3433{)}^2}{216} \\ =54562.44907 \end{gathered}$$

$SSE=SST-SSR$

$$\begin{gathered} SSE=79444.88889-54562.44907 \\ =24882.43981 \end{gathered}$$

The formula of standard error is

$$\begin{gathered} s_e=\sqrt{\frac{SSE}{n-2}} \\ =\sqrt{\frac{24882.43981}{9-2}} \\ ≈59.621 \end{gathered}$$

STEP-3 The formula of test stastic is

$$\begin{gathered} t=\frac{b_1}{s_e/\sqrt{S_{xx}}} \\ =\frac{15.89351852}{59.6207536/\sqrt{216}} \\ ≈3.92 \end{gathered}$$

STEP-4

$$\begin{gathered} df=n-1 \\ =9-2 \\ =7 \end{gathered}$$

The critical values are $Â\pm t_{\mathrm{Î\pm }/2}=Â\pm t_{0.005}=Â\pm 3.500$, as determined by technology.

Or STEP 4: The test statistic's value is $t=3.92$as of Step 3. The P-value is the probability of seeing a value of$tof3.92$or larger in magnitude if the null hypothesis is true, because the test is two-tailed. We derive the $p=00.5765$using technology.

STEP 5: The test statistic's value is smaller than the critical value. Our null hypothesis$H_0$$t=3.92<0.005,7=3.500$, i.e. is rejected. Our null hypothesis$H_0$is not rejected.

STEP 6: The data give sufficient evidence at the 1% significance level to infer that the slope of the population regression line is not $0,$and so the variable $\left(x\right)$is significant.